Why are $K_5$ and $K_{3,3}$ the 'cutoffs' for planarity? I've seen the proof that $K_5$ and $K_{3,3}$ cannot be planar, but I'm curious: is there a why for 5 to be the last complete graph? 
I have to be honest here, I know very little about Graph Theory. I am trying to solve a different problem and it seems that thinking about the problem in terms of planar and complete graphs could be helpful. I searched around for an answer but I couldn't find one. 
I think my question has two possible answers; either it is somewhat analog to asking 'Why are prime numbers infinite?', in the sense that the answer is something like 'They just are.'; or it falls somewhere along the lines of 'Why is the determinant the way it is?', which offers itself to many different types of answers, ranging from sophisticated arguments to more intuitive or geometric ones. I'm hoping my question is more like the latter. 
 A: There are two such "cutoffs" for planarity, called the Kuratowski forbidden minors.
One of those is $K_5$, a pentagram inscribed in a pentagon.  It is not planar.
The other is $K_{3,3}$, the complete bipartite graph in which each of the two parts has three vertices.  It is also not planar.
Here's a harder theorem than the fact that the two graphs above are not planar: There are no others – no other "forbidden minors" for the class of planar graphs.  More precisely: If a graph has no minor isomorphic to either one of these two, then it is planar. This is Kuratowski's theorem. (PS: I find it asserted that that is called Wagner's theorem and that Kuratowksi's theorem is actually stronger than that.)
A minor of a graph $G$ is a a graph that can be made from $G$ by deleting or contracting edges.
The Robertson–Seymour theorem says that every class of graphs that is "downwardly closed", i.e. closed under the formation of minors, is precisely a class of graphs that avoids some finite set of forbidden minors.  Planar graphs are a downwardly closed class of graphs.  Outer-planar graphs (those embeddable in the plane with all vertices on a common circle) are another.  Graphs knotlessly embeddable in $\mathbb R^3$ are another. Graphs linklessly embeddable in $\mathbb R^3$ are another.  Graphs embeddable on a torus are another.
